Let \(P_i\)  denote the prefix of the stream \(\lbrace p_1, \ldots , p_i\rbrace \) . The entire stream is then \(P_n\) . Consider the moment when the first \(i\)  points have arrived, meaning that the prefix \(P_i\)  is the current set of arrived points.
The algorithm \(\mathcal {A}\)  of [1]} provides an \((O(1), O(\log n))\) -approximation of \(P_i\)  in the following sense.
Define \(f_0 : \emptyset \rightarrow \emptyset \)  as the null map, and define \(B_i = \text{image}(f_i)\) . Upon receiving point \(p_i\) , algorithm \(\mathcal {A}\)  defines a map \(f_i : B_{i-1} \cup \lbrace p_i\rbrace  \rightarrow B_i\) . We define \(\pi _i : P_i \rightarrow B_i\)  by \(\pi _i(p_j) = f_i(f_{i-1}(\ldots (f_j(p_j)) \ldots ))\)  for each \(1 \le j \le i\) .
These mappings have an essential gaurantee stated in the following lemma.
